Phase and frequency locking in detailed neuron models R. Stoop, J. Buchli, and M. Christen Institute for Neuroinformatics ETHZ/UNIZH, Winterthurerstr. 190, 8057 Z¨ urich, Switzerland E-mail: ruedi@ini.phys.ethz.ch Abstract—Phase and frequency locking are generic phenomena among coupled limit cycles. As yet, it has been shown that simple models of the neurons (such as the Hodgkin-Huxley and Morris-Lecar) are on limit-cycle so- lutions. The same holds for in vitro pyramidal neurons, if they are driven by constant currents. It is, however, un- known whether this also holds for detailed neuronal com- partment models. In this contribution we show computa- tionally that locking holds and is measurable in these mod- els as well, and can persist even under substantial changes of the driving. Thus, locking among neurons could pro- vide a mechanism for cortical information processing, in the frequency as well as in the temporal coding paradigm. 1. Introduction: Detailed vs. generic neuron models Both experiments and theoretical studies have shown the need to adopt a multi-level approach to understanding the brain, as molecular and genetic events can affect the entire central nervous system, and vice versa. At one extreme, there are the biophysical models of single neurons. These generally are based on one single concept, from which ac- curate neuron models can be constructed: Compartments, i.e., small cylindrical segments of the neuron, are mod- eled with their own complement of ion channels, com- putationally represented by an electrical circuit that takes into account membrane resistance, capacitance, and ionic conductances [1]. Within each compartment, ion currents are typically described as variable conductances in series with the ionic reversal potential. In the extended Hodgkin– Huxley formulation, Ca 2+ -dependent voltage-gated chan- nels generally require multiple state variables as well as Ca- concentration information. From the compartments, indi- vidual neurons can be constructed, and assembled towards full neural networks by means of synaptic conductive inter- connections. On this level, synaptic currents are generally modelled as simple alpha-function responses to an action potential. However: How much biological detail is needed for such a modeling? It is an open question as to whether a detailed numerical model can be more informative than a simpli- fied analytical description of a cell. Neuronal geometry is a crucial determinant both of electrical properties and of neu- ronal connectivity. Every neuronal type has certain charac- teristic branching patterns, which are nevertheless unique from one cell to another. How does one represent the di- versity of a subclass of neurons, while retaining the distin- guishing features of this subclass? To some extent, global information appears to provide a solution to this problem. But, how does one decide on the connectivity models on this level ? Here, the plethora of answers is even larger. The generation of semi-global rhythms [2], e.g., could be one motivation for connectivity models, but there are certainly much more. This all indi- cates that analytical models might be very useful, if they are able to reflect ”all” basic properties of the neuron. In this contribution, we shall focus on this issue in detail, by show- ing that detailed neuronal models – simulated by means of the NEURON environment [3] – are able to reflect the basic properties of limit cycles, which are phase and frequency locking. 2. Neuronal synchronization In 1657, Christiaan Huygens [4] revolutionized the mea- surement of time by creating the first working pendulum clock. In early 1665, he discovered “.. an odd kind of sym- pathy perceived by him in these watches [two pendulum clocks] suspended by the side of each other.” The pendu- lum clocks swung with exactly the same frequency and 180 degrees out of phase; when the pendulums were disturbed, the antiphase state was restored within a half-hour and per- sisted indefinitely. Huygens deduced that the crucial in- teraction for this effect came from “imperceptible move- ments” of the common frame supporting the two clocks. Figure 1: Huygens’ clocks, jointly suspended from a com- mon construction. 2004 International Symposium on Nonlinear Theory and its Applications (NOLTA2004) Fukuoka, Japan, Nov. 29 - Dec. 3, 2004 43